What are the mathematical prerequisites needed to understand research papers on neural networks?
Knowing you want to focus on the theory, I think that a good choice is Deep Learning Book from Ian Goodfellow et al., which is publicly available. It has three main parts. On the first one the author presents the math/ statistic tools that will be needed to understand the following parts. On the second part, the author explains the current state of the art in Deep Learning and on the last part more advanced topics are introduced. The author also uses references so as to facilitate extra resources to dive more into the theory.
On the other hand I strongly recommend you Google Scholar, there you have plenty of information given by articles/ papers of the current state of the art techniques in the field. There you can also find papers related on what that you mentioned about the Hahn-Banach Theorem.